Abstract and Applied Analysis

Cone Monotonicity: Structure Theorem, Properties, and Comparisons to Other Notions of Monotonicity

Heather A. Van Dyke, Kevin R. Vixie, and Thomas J. Asaki

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In search of a meaningful 2-dimensional analog to monotonicity, we introduce two new definitions and give examples of and discuss the relationship between these definitions and others that we found in the literature.

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Abstr. Appl. Anal., Volume 2013 (2013), Article ID 134751, 8 pages.

First available in Project Euclid: 27 February 2014

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Van Dyke, Heather A.; Vixie, Kevin R.; Asaki, Thomas J. Cone Monotonicity: Structure Theorem, Properties, and Comparisons to Other Notions of Monotonicity. Abstr. Appl. Anal. 2013 (2013), Article ID 134751, 8 pages. doi:10.1155/2013/134751. https://projecteuclid.org/euclid.aaa/1393511942

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  • H. Lebesgue, “Sur le problème de Dirichlet,” Rendiconti del Circolo Matematico di Palermo, vol. 27, pp. 371–402, 1907.
  • G. D. Mostow, “Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms,” Institut des Hautes Études Scientifiques. Publications Mathématiques, no. 34, pp. 53–104, 1968.
  • S. K. Vodopyanov and V. M. Goldstein, “Quasiconformal mappings and spaces of functions with generalized first derivatives,” Siberian Mathematical Journal, vol. 17, pp. 399–411, 1976.
  • J. J. Manfredi, “Weakly monotone functions,” The Journal of Geometric Analysis, vol. 4, no. 3, pp. 393–402, 1994.
  • S. G. Krantz and H. R. Parks, Geometric Integration Theory, Birkhäuser Boston Inc., Boston, Mass, USA, 2008.
  • L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1992.